Trigonometry Examples

\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - \sec^{2} (x) + \tan^{2} (x)

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - \sec^{2} (x) + \tan^{2} (x)$

Step 1

Simplify with factoring out.

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Step 1.1

Move

\tan^{2} (x)

$\tan^{2} (x)$ .

\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - \sec^{2} (x) + \tan^{2} (x)

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - \sec^{2} (x) + \tan^{2} (x)$

Step 1.2

Factor

- 1

$- 1$ out of

- \sec^{2} (x)

$- \sec^{2} (x)$ .

\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x)) + \tan^{2} (x)

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x)) + \tan^{2} (x)$

Step 1.3

Factor

- 1

$- 1$ out of

\tan^{2} (x)

$\tan^{2} (x)$ .

\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x)) - 1 (- \tan^{2} (x))

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x)) - 1 (- \tan^{2} (x))$

Step 1.4

Factor

- 1

$- 1$ out of

- (\sec^{2} (x)) - 1 (- \tan^{2} (x))

$- (\sec^{2} (x)) - 1 (- \tan^{2} (x))$ .

\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x) - \tan^{2} (x))

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x) - \tan^{2} (x))$

\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x) - \tan^{2} (x))

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - (\sec^{2} (x) - \tan^{2} (x))$

Step 2

Apply pythagorean identity.

\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - 1 \cdot 1

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} - 1 \cdot 1$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.1.1.1

Add parentheses.

\sin (x) + 3 (\tan (x) \cot (x)) - \frac{1}{\csc (x)} - 1 \cdot 1

$\sin (x) + 3 (\tan (x) \cot (x)) - \frac{1}{\csc (x)} - 1 \cdot 1$

Step 3.1.1.2

Reorder

\tan (x)

$\tan (x)$ and

\cot (x)

$\cot (x)$ .

\sin (x) + 3 (\cot (x) \tan (x)) - \frac{1}{\csc (x)} - 1 \cdot 1

$\sin (x) + 3 (\cot (x) \tan (x)) - \frac{1}{\csc (x)} - 1 \cdot 1$

Step 3.1.1.3

Rewrite

3 \tan (x) \cot (x)

$3 \tan (x) \cot (x)$ in terms of sines and cosines.

\sin (x) + 3 (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)}) - \frac{1}{\csc (x)} - 1 \cdot 1

$\sin (x) + 3 (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)}) - \frac{1}{\csc (x)} - 1 \cdot 1$

Step 3.1.1.4

Cancel the common factors.

\sin (x) + 3 \cdot 1 - \frac{1}{\csc (x)} - 1 \cdot 1

$\sin (x) + 3 \cdot 1 - \frac{1}{\csc (x)} - 1 \cdot 1$

\sin (x) + 3 \cdot 1 - \frac{1}{\csc (x)} - 1 \cdot 1

$\sin (x) + 3 \cdot 1 - \frac{1}{\csc (x)} - 1 \cdot 1$

Step 3.1.2

Multiply

3

$3$ by

1

$1$ .

\sin (x) + 3 - \frac{1}{\csc (x)} - 1 \cdot 1

$\sin (x) + 3 - \frac{1}{\csc (x)} - 1 \cdot 1$

Step 3.1.3

Rewrite

\csc (x)

$\csc (x)$ in terms of sines and cosines.

\sin (x) + 3 - \frac{1}{\frac{1}{\sin (x)}} - 1 \cdot 1

$\sin (x) + 3 - \frac{1}{\frac{1}{\sin (x)}} - 1 \cdot 1$

Step 3.1.4

Multiply by the reciprocal of the fraction to divide by

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ .

\sin (x) + 3 - (1 \sin (x)) - 1 \cdot 1

$\sin (x) + 3 - (1 \sin (x)) - 1 \cdot 1$

Step 3.1.5

Multiply

\sin (x)

$\sin (x)$ by

1

$1$ .

\sin (x) + 3 - \sin (x) - 1 \cdot 1

$\sin (x) + 3 - \sin (x) - 1 \cdot 1$

Step 3.1.6

Multiply

- 1

$- 1$ by

1

$1$ .

\sin (x) + 3 - \sin (x) - 1

$\sin (x) + 3 - \sin (x) - 1$

\sin (x) + 3 - \sin (x) - 1

$\sin (x) + 3 - \sin (x) - 1$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Combine the opposite terms in

\sin (x) + 3 - \sin (x) - 1

$\sin (x) + 3 - \sin (x) - 1$ .

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Step 3.2.1.1

Subtract

\sin (x)

$\sin (x)$ from

\sin (x)

$\sin (x)$ .

0 + 3 - 1

$0 + 3 - 1$

Step 3.2.1.2

Add

0

$0$ and

3

$3$ .

3 - 1

$3 - 1$

3 - 1

$3 - 1$

Step 3.2.2

Subtract

1

$1$ from

3

$3$ .

2

$2$

2

$2$

2

$2$